A device of this kind is known from DE 44 05 332 A1. In this known array, between the exit of the dryer and a chilling roller stand situated immediately downstream of it, provision is made for a tunnel housing through which the paper web passes and which presents two sections arrayed one after the other, of which the section more distant from the dryer takes the form of a remoistening apparatus with chambers containing spray jets, set above and below the transport surface of the paper web, through which moisturising means can be admitted, while the section closer to the dryer is equipped with air squeegees arrayed above and below the transport surface of the paper web, through which compressed air can be admitted, whose function is to scrape off solvents that have been transported together with the paper web. The tunnel housing built onto the dryer constitutes a comparably large volume between the dryer and the chilling roller stand. The cooling zone of the known dryer generally includes air jets through which cool air can be admitted. As the air's heating capacity is relatively small, a relatively large volume is also required in this case, quite regardless of the construction outlay.